U.S. patent number 5,442,442 [Application Number 07/114,481] was granted by the patent office on 1995-08-15 for ring laser gyroscope scale factor error control apparatus and method control apparatus and method.
This patent grant is currently assigned to Litton Systems, Inc.. Invention is credited to Steven C. Gillespie, Edward Kanegsberg, John P. Rahn.
United States Patent |
5,442,442 |
Kanegsberg , et al. |
August 15, 1995 |
**Please see images for:
( Certificate of Correction ) ** |
Ring laser gyroscope scale factor error control apparatus and
method control apparatus and method
Abstract
The intensity and frequency variation due to retroscatter in a
ring laser gyroscope are determined and used to correct the gyro
scale factor. The orthogonal types of scatter due to dielectric
variation and due to height variation, which lead to common mode
phase delays of 0 and .pi./2 respectively are taken into account in
calculating the correction to the scale factor. The scale factor
errors are determined in terms of observable quantities. Scale
factor error control is accomplished by extracting a portion of
both of the two counterpropagating light beams and measuring their
respective intensities, creating intensity modulation indices
representative of the sum and difference intensities, using closed
loop control of the real-time difference between the intensities of
the beam in the ring laser gyro to reduce scale factor variation
using push-pull mirror control of at least two mirrors. The
residual error after push-pull mirror control minimization is
output for use by a navigation system computer.
Inventors: |
Kanegsberg; Edward (Pacific
Palisades, CA), Gillespie; Steven C. (Canoga Park, CA),
Rahn; John P. (Canoga Park, CA) |
Assignee: |
Litton Systems, Inc. (Beverly
Hills, CA)
|
Family
ID: |
22355488 |
Appl.
No.: |
07/114,481 |
Filed: |
October 28, 1987 |
Current U.S.
Class: |
356/473;
372/94 |
Current CPC
Class: |
G01C
19/66 (20130101) |
Current International
Class: |
G01C
19/66 (20060101); G01C 19/64 (20060101); G01B
008/02 (); H01S 003/083 () |
Field of
Search: |
;356/350 ;372/94 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
|
|
|
|
|
|
|
2271542 |
|
Dec 1973 |
|
FR |
|
2749157 |
|
Nov 1978 |
|
DE |
|
1237663 |
|
May 1968 |
|
GB |
|
Other References
Coccoli, "An Overview Of Laser Gyros", 12th Joint Services Data
Exchange for Inertial Systems, Norfolk, Va. 1978. .
Menegozzi et al., "Theory Of A Ring Laser", 1973, Physical Review,
vol. 8, No. 4, pp. 2103-2125. .
Kilpatrick, "The Laser Gyro", Oct. 1967, IEEE Spectrum, pp. 44-55.
.
Aronowitz et al. "Positive Scale Factor Correction in the Laser
Gyro", IEEE Journal of Quantum Electronics, vol. QE-13, No. 5 May
1977, pp. 338-343. .
Hammons et al. "Mechanically Dithered RLG at the Quantum Limit",
1982 IEEE NAECON, pp. 388-392..
|
Primary Examiner: Buczinski; Stephen C.
Attorney, Agent or Firm: Lynn & Lynn
Claims
What is claimed is:
1. A scale factor error control system for a ring laser gyroscope
that includes a frame having a cavity therein for guiding a pair of
counterpropagating light beams and a plurality of cavity length
control mirrors mounted to the frame comprising:
means for extracting a portion of each of the two
counterpropagating light beams from the cavity;
means for forming a first beam intensity signal indicative of the
intensity of the portion of a first one of the two
counterpropagating beams extracted from the cavity and a second
beam intensity signal indicative of the intensity of the portion of
the other one of the two counterpropagating beams extracted from
the cavity, each beam intensity signal having an AC component and a
DC component;
means for forming intensity modulation indices representative of
the sum and difference of the first and second beam intensity
signals, including;
means for forming a first ratio by dividing a first signal
indicative of the AC component of the first beam intensity signal
by a second signal indicative of the DC component of the first beam
intensity signal; and
means for forming a second ratio by dividing a second signal
indicative of the AC component of the second beam intensity signal
by a second signal indicative of the DC component of the second
beam intensity signal; and
means for minimizing the difference and sum of the first and second
beam intensity signals.
2. The scale factor error control system of claim 1, wherein the
means for minimizing the difference and sum of the first and second
beam intensity signals further includes means for providing
push-pull mirror control of at least two of the cavity length
control mirrors.
3. The scale factor error control system of claim 1 wherein the
extracting means further comprises:
a partially transmissive mirror mounted to the frame to receive the
counterpropagating beams thereon so that portions of each beam
propagate out of the cavity;
means responsive to a first beam that has propagated from the
cavity for producing the first beam intensity signal; and
means responsive to a second beam that has propagated from the
cavity for producing the second beam intensity signal.
4. The scale factor control system of claim 1 further
comprising:
first photodetector means for producing the first beam intensity
signal;
second photodetector means for producing the second beam intensity
signal;
means for combining signals output from the first and second
photodetector means; and
means for processing signals indicative of the difference between
the intensities of the counterpropagating light beams and signals
output from the means for combining signals output from the first
and second photodetector means to reduce scale factor error and
maintain constant cavity length.
5. The scale factor control system of claim 1 wherein the means for
minimizing the difference between the differences and sums of the
beam intensities comprises:
means for producing an electrical signal indicative of the
displacement of a mirror away from its point of minimum scale
factor error;
means for applying the electrical signal to a first piezoelectric
actuator to move a first mirror into a location providing minimum
scale factor error; and
means for applying the electrical signal to a second piezoelectric
actuator to move a second mirror into a location providing minimum
scale factor error.
6. A scale factor error control system for a ring laser gyroscope
that includes a frame having a cavity therein for guiding a pair of
counterpropagating light beams and a plurality of cavity length
control mirrors mounted to the frame comprising:
means for extracting a portion of each of the two
counterpropagating light beams from the cavity, the extracting
means including a partially transmissive mirror mounted to the
frame to receive the counterpropagating beams thereon so that
portions of each beam propagate out of the cavity;
means for measuring a first beam intensity for the portion of a
first one of the two counterpropagating beams extracted from the
cavity and a second beam intensity for the portion of the other one
of the two counterpropagating beams extracted from the cavity;
means for forming intensity modulation indices representative of
the sum end difference of the first and second beam intensities,
including:
means for producing an amplified AC component of the first beam
intensity signal;
means for producing an amplified AC component of the second beam
intensity signal;
means for producing an amplified DC component of the first beam
intensity signal;
means for producing an amplified DC component of the second beam
intensity signal;
means for dividing the amplified AC component of the first beam
intensity signal by the amplified DC component of the first beam
intensity signal to form a first ratio; and
means for dividing the amplified AC component of the second beam
intensity signal by the amplified DC component of the second beam
intensity to form a second ratio; and
means for minimizing the differences and sums of the first and
second beam intensities.
7. The scale factor control system of claim 6 wherein the means for
forming intensity modulation indices comprises:
means for determining the difference between the first ratio and
the second ratio; and
means for summing the first ratio and the second ratio.
8. The scale factor error control system of claim 7, wherein the
means for forming intensity modulation indices comprises:
means for dual demodulating signals output from the means for
determining the difference between the first ratio and the second
ratio using signals indicative of the beat frequency between the
waves in the cavity as reference signals; and
means for dual demodulating signals output from the means for
summing the first ratio and the second ratio using the signals
indicative of the beat frequency between the waves in the cavity as
reference signals.
9. The scale factor control system of claim 7, further including
demodulating means comprised of a pair of dual phase locked loop
demodulators.
10. The scale factor error control system of claim 6 wherein the
means for forming intensity modulation indices includes:
means for determining the difference between the first ratio and
the second ratio;
first low pass filtering means for low pass filtering signals
output from the means for determining the difference between the
first ratio and the second ratio;
means for summing the first ratio and the second ratio;
second low pass filtering means for low pass filtering signals
output from the means for summing the first ratio and the second
ratio;
means for converting signals output from the first low pass
filtering means from analog to digital; and
means for converting signals output from the second low pass
filtering means from analog to a digital signal.
11. The scale factor error control system of claim 6 including
signal processing and arithmetic-logic means that comprises:
system computing means for computing scale factors;
means for producing a signal indicative of the rotation rate of the
ring laser gyroscope;
means for producing a signal indicative of the intensity of a light
beam propagating within the ring laser gyroscope in the counter
clockwise direction;
means for producing a signal indicative of the intensity of a light
beam propagating within a ring laser gyroscope in the clockwise
direction; and
means for combining the signal indicative of the rotation rate and
the signals indicative of the light beam intensities to generate a
scale factor error correction for input to the system computing
means to be subtracted from the scale factor used therein.
12. A method of scale factor error control for a ring laser
gyroscope that includes a frame having a cavity therein for guiding
a pair of counterpropagating light beams and two cavity length
control mirrors mounted to the frame, comprising the steps of:
extracting a portion of each of the two counterpropagating light
beams from the cavity;
forming a first beam intensity signal indicative of the intensity
of the portion of a first one of the two counterpropagating beams
extracted from the cavity;
forming a second beam intensity signal indicative of the intensity
of the portion of the other one of the two counterpropagating beams
extracted from the cavity, each beam intensity signal having an AC
component and a DC component,
forming intensity modulation indices representative of the sum and
difference of the first and second beam intensity signals,
including the steps of:
forming a first ratio by dividing a first signal indicative of the
AC component of the first beam intensity signal by a second signal
indicative of the DC component of the first beam intensity signal;
and
forming a second ratio by dividing a second signal indicative of
the AC component of the second beam intensity signal by a second
signal indicative of the DC component of the second beam intensity
signal; and
using closed loop control of the difference between the intensities
of the first and second beam intensity signals to minimize the
difference and sum of the first and second beam intensity signals
to reduce scale factor variation.
13. The method of claim 12 including the step of providing
push-pull mirror control of at least two mirrors.
14. The method of claim 12 wherein the extracting step further
comprises the steps of:
producing a signal indicative of the intensity of the first
extracted light wave; and
producing a signal indicative of the intensity of the second
extracted light wave.
15. The method of claim 12 wherein the step of forming intensity
modulation indices further comprises the steps of:
determining the difference between the first ratio and the second
ratio; and
summing the first ratio with the second ratio.
16. The method of claim 12, further comprising the steps of:
producing a signal indicative of the rotation rate of the ring
laser gyroscope;
producing a signal indicative of the intensity of a light beam
propagating within a ring laser gyroscope in the counter clockwise
direction;
producing a signal indicative of the intensity of a light beam
propagating within a ring laser gyroscope in the clockwise
direction;
processing the beam intensity signals to generate scale factor
error correction.
17. The method of claim 12, further comprising the steps of:
combining the first and second beam intensity signals; and
processing the difference between the intensities of the
counterpropagating light beams and the result of combining the
first and second beam intensity signals to reduce scale factor
error and maintain constant cavity length.
18. The method of claim 12 wherein the step of using closed loop
control of the difference between the intensities of the first and
second beam intensity signals to minimize the difference between
light beam intensities using push-pull mirrors further comprises
the steps of:
producing a first electrical signal indicative of the displacement
of a first cavity length control mirror away from its point of
minimum scale factor error;
producing a second an electrical signal indicative of the
displacement of a second cavity length control mirror away from its
point of minimum scale factor error;
connecting the first electrical signal to a first piezoelectric
actuator to move the first mirror into a location providing minimum
scale factor error; and
connecting the second electrical signal to a second piezoelectric
actuator to move a second mirror into a location providing minimum
scale factor error.
19. The method of claim 12, wherein the step of forming intensity
modulation indices further comprises the steps of:
dual demodulating signals indicative of the difference between the
light beam intensities using signals indicative of the beat
frequency between the waves in the cavity as reference signals;
and
dual demodulating signals indicative of the sum of the light beam
intensities using signals indicative of the beat frequency between
the waves in the cavity as reference signals.
20. A method of scale factor error control for a ring laser
gyroscope that includes a frame having a cavity therein for guiding
a pair of counterpropagating light beams and two cavity length
control mirrors mounted to the frame, comprising the steps of:
extracting a portion of each of the two counterpropagating light
beams;
forming a first beam intensity signal indicative of the intensity
of the portion of a first one of the two counterpropagating beams
extracted from the cavity;
forming a second beam intensity signal indicative of the intensity
of the portion of the other one of the two counterpropagating beams
extracted from the cavity, each beam intensity signal having an AC
component and a DC component,
forming intensity modulation indices representative of the sum and
difference of the first and second beam intensity signals,
including the steps of:
forming a first ratio by dividing a first signal indicative of the
AC component of the first beam intensity signal by a second signal
indicative of the DC component of the first beam intensity
signal;
forming a second ratio by dividing a second signal indicative of
the AC component of the second beam intensity signal by a second
signal indicative of the DC component of the second beam intensity
signal;
dual demodulating of signals indicative of the difference between
the light beam intensities using signals indicative of the beat
frequency between the waves in the cavity as reference signals;
and
dual demodulation of signals indicative of the sum of the first and
second beam intensity signals using a signal indicative of the beat
frequency between the waves in the cavity as a reference signal;
and
using closed loop control of the difference between the intensities
of the first and second beam intensity signals to minimize the
difference and sum of the first and second beam intensity signals
to reduce scale factor variation.
21. The method of claim 20 wherein the step of forming intensity
modulation indices further includes the steps of:
low pass filtering with a first low pass filter the dual
demodulated difference between the first and second beam intensity
signals;
low pass filtering with a second low pass filter the dual
demodulated sum of the first and second beam intensity signals;
converting signals output from the first low pass filter from
analog to digital; and
converting signals output from the second low pass filter from
analog to digital.
22. A ring laser gyroscope that includes a frame having a cavity
therein for guiding a pair of counterpropagating light beams and a
plurality of cavity length control mirrors mounted to the frame
comprising:
means for extracting a portion of each of the two
counterpropagating light beams from the cavity and forming signals
indicative of the intensities of the beams, including:
a partially transmissive mirror mounted to the frame to receive the
counterpropagating beams thereon so that portions of each beam
propagate out of the cavity;
means responsive to a first beam that has propagated from the
cavity for producing a first beam intensity signal; and
means responsive to a second beam that has propagated from the
cavity for producing a second beam intensity signal.;
means for forming intensity modulation indices representative of
the sum and difference of the first and second beam intensities,
including:
means for producing an amplified AC component of the first beam
intensity signal;
means for producing an amplified AC component of the second beam
intensity signal;
means for producing an amplified DC component of the first beam
intensity signal;
means for producing an amplified DC component of the second beam
intensity signal;
means for dividing the amplified AC component of the first beam
intensity signal by the amplified DC component of the first beam
intensity signal to form a first ratio; and
means for dividing the amplified AC component of the second beam
intensity signal by the amplified DC component of the second beam
intensity to form a second ratio; and
means for forming a first ratio by dividing a first signal
indicative of the AC component of the first beam intensity signal
by a second signal indicative of the DC component of the first beam
intensity signal; and
means for forming a second ratio by dividing a second signal
indicative of the AC component of the second beam intensity signal
by a second signal indicative of the DC component of the second
beam intensity signal; and
means for forming a pair of heterodyne signals that are indicative
of the beat frequency of the counterpropagating light beams in the
cavity, the heterodyne signals being related to the rotation rate
of the frame by a scale factor; and
means for processing the beam intensity signals and the heterodyne
signals to provide a scale factor correction signal to be used in
determining the rotation rate of the frame from the heterodyne
signals.
23. The ring laser gyroscope of claim 22 wherein the means for
forming intensity modulation indices comprises:
means for determining the difference between the first ratio and
the second ratio; and
means for summing the first ratio and the second ratio.
24. The ring laser gyroscope of claim 23, wherein the means for
forming intensity modulation indices comprises:
means for dual demodulating the difference between the first ratio
and the second ratio using signals indicative of the beat frequency
between the waves in the cavity as reference signals; and
means for dual demodulating the sum of the first ratio and the
second ratio using the signals indicative of the beat frequency
between the waves in the cavity as reference signals.
25. A method of measuring rotations with a ring laser gyroscope
that includes a frame having a cavity therein for guiding a pair of
counterpropagating light beams and a plurality of cavity length
control mirrors mounted to the frame, comprising the steps of:
extracting a portion of each of the two counterpropagating light
beams from the cavity and forming signals indicative of the
intensities of the beams by the steps of:
mounting a partially transmissive mirror to the frame to receive
the counterpropagating beams thereon so that portions of each beam
propagate out of the cavity;
producing a first beam intensity signal responsive to a first beam
that has propagated from the cavity; and
producing a second beam intensity signal responsive to a second
beam that has propagated from the cavity;
forming intensity modulation indices representative of the sum and
difference of the first and second beam intensities, including:
producing an amplified AC component of the first beam intensity
signal;
producing an amplified AC component of the second beam intensity
signal;
producing an amplified DC component of the first beam intensity
signal;
producing an amplified DC component of the second beam intensity
signal;
dividing the amplified AC component of the first beam intensity
signal by the amplified DC component of the first beam intensity
signal to form a first ratio; and
dividing the amplified AC component of the second beam intensity
signal by the amplified DC component of the second beam intensity
to form a second ratio; and
forming a first ratio by dividing a first signal indicative of the
AC component of the first beam intensity signal by a second signal
indicative of the DC component of the first beam intensity signal;
and
forming a second ratio by dividing a second signal indicative of
the AC component of the second beam intensity signal by a second
signal indicative of the DC component of the second beam intensity
signal; and
forming a pair of heterodyne signals that are indicative of the
beat frequency of the counterpropagating light beams in the cavity,
the heterodyne signals being related to the rotation rate of the
frame by a scale factor; and
processing the beam intensity signals and the heterodyne signals to
provide a scale factor correction signal to be used in determining
the rotation rate of the frame from the heterodyne signals.
26. The method of claim 25 wherein the step of forming intensity
modulation indices comprises the steps of:
determining the difference between the first ratio and the second
ratio; and
summing the first ratio and the second ratio.
27. The method of claim 26, wherein the step of forming intensity
modulation indices comprises the steps of:
forming a signal indicative of the difference between the first and
second beam intensity signals;
dual demodulating the signal indicative of the difference between
the light beam intensities using signals indicative of the beat
frequency between the waves in the cavity as reference signals;
and
forming a signal indicative of the sum of the first and second beam
intensity signals;
dual demodulating the signal indicative of the sum of the first and
second beam intensity signals using a the signals indicative of the
beat frequency between the waves in the cavity as a reference
signals.
28. The method of claim 27, further including the steps of:
forming a signal indicative of the sum of the first and second beam
intensity signals;
forming a signal indicative of the difference between the first and
second beam intensity signals;
demodulating the signal sum of the first and second beam intensity
signals with a first dual phase locked loop demodulator; and
demodulating the signal difference of the first and second beam
intensity signals with a first dual phase locked loop
demodulator.
29. The method of claim 25 including minimizing the difference
between the differences and sums of the beam intensities by the
steps of:
producing an electrical signal indicative of the displacement of a
mirror away from its point of minimum scale factor error;
applying the electrical signal to a first piezoelectric actuator to
move a first mirror into a location providing minimum scale factor
error; and
applying the electrical signal to a second piezoelectric actuator
to move a second mirror into a location providing minimum scale
factor error.
Description
BACKGROUND OF THE INVENTION
This invention relates generally to rotation sensors and
particularly to ring laser gyroscope rotation sensors. Still more
particularly, this invention relates to apparatus and methods for
stabilizing the frequency of light produced in a ring laser.
A ring laser gyroscope employs the Sagnac effect to detect
rotation. Two counterpropagating light beams in a closed loop will
have transit times that differ in direct proportion to the rotation
rate of the loop about an axis perpendicular to the plane of the
loop. There are in general two basic techniques for utilizing the
Sagnac effect to detect rotations. A first technique is the
interferometric approach, which involves measuring the differential
phase shift between two counterpropagating beams injected from an
external source, typically a laser, into a Sagnac ring. The ring
may be defined by mirrors that direct the light beams around the
path or by a coil of optical fiber. Beams exiting the path
interfere and create a pattern of light and dark lines that is
usually called a fringe pattern. Absolute changes in the fringe
pattern are indicative of rotation of the ring. The primary
difficulty with such devices is that the changes are very small for
rotation rates of interest in guidance applications.
The ring laser gyroscope uses the resonant properties of a closed
cavity to convert the Sagnac phase difference between the counter
propagating beams into a frequency difference. The high optical
frequencies of about 10.sup.15 Hz for light used in ring laser
gyroscopes cause the minute phase changes to become beat
frequencies that are readily measured. The cavity length must be
precisely controlled to provide stability in the lasing frequency.
A stable frequency is required to provide the desired accuracy in
measuring rotations.
A ring laser gyroscope has a sensor axis that passes through the
closed paths traversed by the counterpropagating beams. When the
ring laser gyroscope is not rotating about its sensor axis, the
optical paths for the two counterpropagating beams have identical
lengths so that the two beams have identical frequencies. Rotation
of the ring laser gyroscope about its sensor axis causes the
effective path length for light traveling in the direction of
rotation to increase while the effective path length for the wave
traveling in the direction opposite to the rotation decreases.
Ring laser gyroscopes may be classified as passive or active,
depending upon whether the lasing, or gain medium is external or
internal to the cavity. In the active ring laser gyroscope the
cavity defined by the closed optical path becomes an oscillator,
and output beams from the two directions beat together to give a
beat frequency that is a measure of the rotation rate. The
oscillator approach means that the frequency filtering properties
of the cavity resonator are narrowed by many orders of magnitude
below the passive cavity and give very precise rotation sensing
potential. To date the major ring laser gyroscope rotation sensor
effort has been put into the active ring laser. Presently all
commercially available optical rotation sensors are active ring
laser gyroscopes.
When the rotation rate of the ring laser gyroscope is within a
certain range, the frequency difference between the beams
disappears. This phenomenon is called frequency lock-in, or mode
locking, and is a major difficulty with the ring laser gyroscope
because at low rotation rates, the frequency difference between the
beams disappears. This input rotation rate is called the lock-in
threshold. The range of rotation rates over which lock-in occurs is
the deadband of the ring laser gyroscope.
Lock-in is believed to arise from coupling of light between the
beams. The coupling results primarily from backscatter off the
mirrors that confine the beams to the closed path. Backscatter
causes the beam in each direction to include a small component
having the frequency of the beam propagating in the other
direction. The lock-in effect in a ring laser gyroscope is similar
to the coupling that has long been observed and understood in
conventional electronic oscillators.
Upon reversal of the sign of the frequency difference between the
two beams, there is a tendency for the beams to lock-in since at
some point the frequency difference is zero. Since the output of
the ring laser gyroscope is derived from the frequency difference,
an error accumulates in the output angle. The periods in which the
two beams are locked in are usually very short in duration; thus,
the error is very small. However, since the error is cumulative, in
time the error can become appreciable in precision navigation
systems. This error is a major contributor to random walk or random
drift.
In addition to causing erroneous rotation rate information to be
output from a ring laser gyroscope, lock-in causes standing waves
to appear on the mirror surfaces. These standing waves may create a
grating of high and low absorption regions, which create localized
losses that increase the coupling between the beams and the
lock-in. The mirrors may be permanently distorted by leaving a ring
laser gyroscope operating in a lock-in condition.
Any inability to accurately measure low rotation rates reduces the
effectiveness of a ring laser gyroscope in navigational systems.
There has been substantial amount of research and development work
to reduce or eliminate the effects of lock-in to enhance the
effective use of ring laser gyroscopes in such systems.
There are several known approaches to solving the problems of
lock-in. One such approach involves mechanically oscillating the
ring laser gyroscope about its sensor axis so that the device is
constantly sweeping through the deadband and is never locked
therein. This mechanical oscillation of the ring laser gyroscope is
usually called dithering.
In one implementation of the RLG, lock-in is largely subdued by
placing a sinusoidal plus random mechanical rotational dither of
about 200 arc seconds amplitude and at a rate of a few hundred
Hertz on the input axis of the RLG. This greatly reduces the scale
factor nonlinearity but does not completely eliminate it.. The
present invention seeks to minimize scale factor variations and
nonlinearities by controlling the scatter and by providing a
discriminant for correction of scale factor error.
Basically, there are two mirror scatterer types: that due to
surface roughness or height variation, H, which induces a common
mode phase shift of 90 degrees, .pi./2 radians, and that due to
dielectric variations, D, which induces a phase shift of 0 degrees.
These 2 scattering sources also have asymmetrical phase shifts
which result in the clockwise (cw) traveling scattered field
appearing as the complex conjugate of the counter-clockwise (ccw)
scattered field.
Previous methods of scale factor variation reduction involved
reducing scatter and increasing dither amplitude. These have
distinct limitations in terms of cost and mechanical complexity.
The present invention attempts to reduce scale factor nonlinearity
by an ensemble average of a factor of 4 and may reduce the
variations by a much larger factor, all for a moderate cost and
moderate increase in complexity.
SUMMARY OF THE INVENTION
In reducing the scale factor error in a ring laser gyroscope this
invention concentrates on computing the intensity and frequency
variation due to retroscatter in the ring laser gyro. The invention
incorporates the resonator leg lengths and the mirror scatter
asymmetry into the phasor addition of the scatter from RLG mirrors.
This invention also takes into account the two orthogonal types of
scatter, either that due to dielectric variation or that due to
height variation, which lead to common mode phase shifts of 0 and
.pi./2 respectively. The lockin rate and scale factor errors are
determined in terms of experimentally observable quantities.
The apparatus of the present invention comprises a scale factor
error control system for a ring laser gyroscope that includes a
frame having a cavity therein for guiding a pair of
counterpropagating light beams and two cavity length control mirror
mounted to the frame. The apparatus includes means for extracting a
portion of both of the two counterpropagating light beams and
measuring their respective intensities, means for creating maximum
intensity modulation indices representative of the sum and
difference intensities, means for providing signal processing and
arithmetic logic functions, means for using closed loop control of
the real-time difference between the intensities of the beam in the
ring laser gyro to reduce scale factor variation, and means for
minimizing the difference between light beam intensities using
push-pull mirror control of at least two mirrors.
The present invention may also comprise means for producing a
signal indicative of the intensity of the first counterpropagating
light wave and means for producing a signal indicative of the
intensity of the second counterpropagating light wave. The present
invention may also include means for amplifying the AC component of
the signal produced by a first photodetector, means for amplifying
the AC component of the signal produced by a second photodetector,
means for amplifying the DC component of the signal produced by the
first photodetector means, means for amplifying the DC component of
the signal produced by the second photodetector, means for dividing
the first amplified AC component by the first amplified DC
component to form a first ratio, and means for dividing the second
amplified AC component by the second amplified DC component to form
a second ratio.
The apparatus of the present invention preferably comprises means
for determining the difference between the first ratio and the
second ratio and means for summing the first ratio with the second
ratio. The apparatus may include means for dual demodulation of the
difference of the two ratios using signals from the first and
second photodetector means as reference signals, and means for dual
demodulation of the sum of the two ratios using the signals from
the first and second photodetectors as reference signals.
The present invention includes means for producing a signal
indicative of the rotation rate of the ring laser gyroscope, means
for producing a signal indicative of the rotation rate of a light
beam propagating within a ring laser gyroscope in the counter
clockwise direction, means for producing a signal indicative of the
rotation rate of a light beam propagating within a ring laser
gyroscope in the clockwise direction, means for computing an analog
quantity involving the sums and differences of AC to DC ratios, and
means for generating a scale factor error correction for output to
the system computer to be subtracted out of the scale factor error
used therein.
The apparatus of the present invention includes means for combining
the outputs of the first and second photodetector means, and means
for using the difference between the intensities of the
counterpropagating light beams and the combined output to reduce
scale factor error and maintain constant cavity length.
The present invention includes means for producing an electrical
signal indicative of the displacement of a mirror away from its
point of minimum scale factor error, means for connecting the
indicative signal to a first actuator to move the mirror into a
location providing minimum scale factor error, and means for
connecting the electrical signal to a second actuator to move a
second mirror into a location providing minimum scale factor
error.
The method of the present invention includes the steps of providing
scale factor error control for a ring laser gyroscope that includes
a frame having a cavity therein for guiding a pair of
counterpropagating light beams and two cavity length control mirror
mounted to the frame. The method includes the steps of extracting a
portion of both of the two counterpropagating light beams and
measuring their respective intensities, creating maximum intensity
modulation indices representative of the sum and difference
intensities, providing signal processing and arithmetic logic
functions, using closed loop control of the real-time difference
between the intensities of the beam in the ring laser gyro to
reduce scale factor variation, and minimizing the difference
between light beam intensities using push-pull mirror control of at
least two mirrors.
The present invention may also comprise the steps of producing a
signal indicative of the intensity of the first counterpropagating
light wave and producing a signal indicative of the intensity of
the second counterpropagating light wave. The present invention may
further include the steps of amplifying the AC component of the
signal produced by a first photodetector, amplifying the AC
component of the signal produced by a second photodetector,
amplifying the DC component of the signal produced by the first
photodetector means, amplifying the DC component of the signal
produced by the second photodetector, dividing the first amplified
AC component by the first amplified DC component to form a first
ratio, and dividing the second amplified AC component by the second
amplified DC component to form a second ratio. The method of the
invention comprises the steps of determining the difference between
the first ratio and the second ratio and summing the first ratio
with the second ratio.
The method of the present invention includes the steps of providing
dual demodulation of the output of the difference of the two ratios
using signals from the first and second photodetector means as
reference signals, and dual demodulation of the sum of the two
ratios using the signals from the first and second photodetector
means as reference signals.
The method of the present invention further comprises the steps of
producing a signal indicative of the rotation rate of the ring
laser gyroscope, producing a signal indicative of the rotation rate
of a light beam propagating within a ring laser gyroscope in the
counter clockwise direction, producing a signal indicative of the
rotation rate of a light beam propagating within a ring laser
gyroscope in the clockwise direction, computing an analog quantity
involving the sums and differences of the AC to DC ratios, and
generating a scale factor error correction for output to the system
computer to be subtracted out of the scale factor error used
therein.
The present invention comprises the steps of combining the outputs
of the first and second photodetector means and using the
difference between the intensities of the counterpropagating light
beams and the combined output to reduce scale factor error and
maintain constant cavity length.
The present invention includes the steps of producing an electrical
signal indicative of the displacement of a mirror away from its
point of minimum scale factor error, connecting the indicative
signal to a first piezoelectric unit to move a first mirror into a
location providing minimum scale factor error, and connecting the
indicative electrical signal to a second piezoelectric unit to move
a second mirror into a location providing minimum scale factor
error.
DESCRIPTION OF THE DRAWINGS
FIG. 1 is a perspective view of a ring laser gyroscope;
FIG. 2 illustrates apparatus for dithering a mirror in the ring
laser gyroscope of FIG. 1;
FIG. 3 schematically illustrates the apparatus of the present
invention for scale factor error control of the ring laser
gyroscope of FIG. 1; and
FIG. 4 depicts circuitry that may be included in a dual phase
demodulator that may be included in the circuit of FIG. 3.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Referring to FIG. 1, a ring laser gyroscope 10 includes a frame 12
that may have a polygonal shape with the corners cut off to form
four mounting faces 14-17. A plurality of mirrors 18-21 are mounted
on the mounting faces 14-17, respectively. A cavity 22 is formed in
the frame 12 to form a path around the frame 12 between the mirrors
18-21.
A lasing medium 23 is positioned in the cavity 22 to produce
counterpropagating light beams therein represented by the arrows 25
and 27 in the lasing medium 23. The lasing medium 23 is typically a
mixture of helium and neon confined to the cavity 22. Energy may be
delivered to the lasing medium 23 by a pair of power supplies 35
and 37 which apply voltage between a pair of anodes 24 and 26 and
to a cathode 28. Other structures may be used for the ring laser
gyroscope 10. The basic description of the ring laser gyroscope 10
is presented by way of example and not for limitation of the
present invention to a particular ring laser gyroscope
structure.
Referring to FIGS. 1 and 3, one of the mirrors, for example the
mirror 20 is partly transmissive so that a portion of each beam
exits the mirror 20. The two counterpropagating beams undergo phase
shifts in circulating around the cavity 22 as the cavity 22 rotates
about its sensor axis. These phase shifts may be viewed as changes
in the frequencies of the two beams. The difference between the
frequencies of the two counterpropagating beams is indicative of
the rotation rate of the cavity 22 about its sensor axis. Since the
cavity 22 acts as a resonant cavity to the two beams, the frequency
of each beam is sharply defined.
FIG. 3 illustrates apparatus according to this invention for
providing scale factor error correction and minimization. Referring
to FIGS. 1 and 3 the mirror 20 is only partly reflective, and it
allows a portion of the laser light to be transmitted out of the
ring laser cavity 22. Referring to FIG. 3, light travels to a first
photodetector 50 and a second photodetector 52. The first
photodetector 50 is positioned in such a way that it detects the
intensity of the light beam propagating in the counter clockwise
direction. The second photodetector 52 is positioned in such a way
that it detects the intensity of the light beam propagating in the
clockwise direction.
The first photodetector 50 produces an electrical signal indicative
of the intensity of the detected counter clockwise propagating
light beam. The second photodetector 52 produces an electrical
signal indicative of the intensity of the detected clockwise
propagating light beam.
The signals output from these two electrical sources 50 and 52 are
fed to several locations in the circuit shown in FIG. 3. The first
is capacitively coupled to an AC preamplifier 54 which separately
amplifies the AC components of both the first and the second
electric signals.
The electrical beam intensity signals are also connected to a DC
preamplifier 56 which separately amplifies the DC component of both
the first and second photodetector output signals. The preamplified
AC signal from the first photodetector 50 is then divided by the
preamplified DC component of the first photodetector output by a
first analog divider 58. The preamplified signal from the second
photodetector 52 is divided by the DC component of the second
photodetector output by a second analog divider 60.
The signal from the first analog divider 58 is added to the signal
from the second analog divider 60 in a summing amplifier 62. The
difference between the first ratio and the second ratio is
determined by the difference amplifier 64. These two signals, the
sum signal and difference signal, have the same frequency as the
photodetector output signals.
The rate output of the ring laser gyroscope 10 may be obtained by
combining the beams in a prism 59 so that they become nearly
parallel. The prism 59 is shown attached to the mirror 19 in FIG.
3. The combined beams impinge on a pair of heterodyne
photodetectors 61 and 63. The combined beams produce an
interference pattern that moves across the heterodyne detectors 61
and 63 as the frame 12 rotates about its sensing axis. The signals
output from photodetectors 61 and 63 referred to as HET A and HET
B, respectively.
The output from the summing amplifier 62 is fed to a first
demodulator 66 which is a dual demodulator described subsequently
with reference to FIG. 4. The signal HET A is used as the first
reference frequency, and the signal HET B is used as the second
reference frequency for demodulating the sum signal. The output of
the difference amplifier 64 is also fed to a dual demodulator 68,
which is similar to the demodulator 66. The reference signals HET A
and HET B are connected to the demodulator 68 in the same
arrangement as for the first dual demodulator 66.
The structure of the dual demodulator 66 is shown in FIG. 4. The
demodulator 68 is identical to the demodulator 66. The part of the
amplitude modulation of the beam intensity caused by backscatter
has the same frequency as the signals HET A and HET B. Therefore,
demodulation with the heterodyne signals (or squares of the same
phase and frequency) as a reference enhances the detection of the
amplitude modulation caused by backscatter. However, the phase of
the intensity signals relative to the heterodyne signals cannot be
predicted. The intensity detectors 50 and 52 are on a different
corner of the frame 12 from the combining optics 59, and since
neither the initially constructed path length nor the path length
stability between these corners can be controlled, no assumption as
to phase can be made. The dual phase demodulator therefore includes
an in-phase demodulator 150 and a quadrature-phase demodulator 152
connected to receive the intensity signal from the summer 62. Since
the two demodulators 150 and 152 are in quadrature with one
another, the intensity signal will always have a component in phase
with at least one of them.
The signal HET A is input to the in-phase demodulator 150. Both of
the HET A and HET B signals are input to a rotation sense logic
circuit 154, which outputs either the HET B signal or its logic
inverse to the quadrature-phase demodulator 152. The output from
the rotation sense circuit 154 to the quadrature-phase demodulator
depends upon the direction of rotation of the frame 12.
The outputs of the in-phase and quadrature phase demodulators 150
and 152 are input to low pass filters 156 and 158, respectively.
The filtered in phase signal may then be input to a processing
circuit 160 that determines the absolute value of the signal. The
filtered quadrature phase signal may be input to a processing
circuit 162 that determines the absolute value of the quadrature
phase signal. A summing circuit 164 then adds the outputs of the
processing circuits to determine the sum of the absolute values of
the filtered signals.
Instead of determining the absolute values of the filtered signals,
the processing circuits 160 and 162 may determine the squares of
the signals. The output of the summing circuit 164 is then the sum
of the squares of the demodulated and filtered in-phase and
quadrature phase signals.
Both of the outputs from the dual demodulators 66 and 68 are low
pass filtered through low pass filters 70 and 72, respectively. The
first and second filtered analog signals are then converted to
digital by two analog-to-digital converters 74 and 76. The sum
signal is converted through the first analog-to-digital converter
74 and the difference signal is converted through the second
analog-to-digital converter converter 76.
The digital output from converter 76 is referred to as m.sub.1,
which is the modulation index of the difference intensity and is a
function of the rotation rate of the ring laser gyroscope. It is a
modulated representation of the difference in intensities of the
counter propagating light beams in a form which can be manipulated
by the system computer 80.
The digital output from converter 74 is referred to as M.sub.1,
which is the modulation index of the sum intensities and is also a
function of ring laser gyroscope rotation rate. It is a modulated
representation of the sum of intensities of the counter propagating
light beams, in a form which can be manipulated by a system
computer 80 that receives the digital signals from the converters
74 and 76.
A quantity m.sub.1.sup.2 -CM.sub.1.sup.2 is computed for the
minimization of scale factor error by mirror movement. The symbol C
represents a quantity that depends upon the input rotation rate as
explained subsequently. The residue of this quantity is input to
the system computer to be subtracted out of the scale factor. This
quantity m.sub.1.sup.2 -CM.sub.1.sup.2 is also used to reduce scale
factor error by push-pull mirror control of at least 2 mirrors.
Push-pull mirror control as used herein means that one mirror is
moved out relative to the frame while the other is moved in by an
equal distance. The mirrors are moved slightly to reduce
m.sub.1.sup.2 -CM.sub.1.sup.2. At least two mirrors must be moved
so that the overall cavity length of the ring laser gyroscope 10
remains constant as the mirrors are moved.
This push-pull mirror control is accomplished through the following
process. The scale factor error mirror and the cavity length
control mirror dither drive unit 104 oscillates the mirrors at very
small amplitudes in push-pull fashion at a frequency substantially
different from the cavity length control mirror drive, which also
oscillates the mirrors in push-pull. The scale factor error summing
circuit 90 receives both the computed function m.sub.1.sup.2
-CM.sub.1.sup.2 from the digital-to-analog converter 88 and a
dither reference signal. The circuit 104 synchronously demodulates
the function m.sub.1.sup.2 -CM.sub.1.sup.2 and the amplified output
of this circuit is added in series in push-pull fashion to the
cavity length control voltage output in a manner the minimizes
m.sub.1.sup.2 -CM.sub.1.sup.2 and the scale factor error. The
quantity m.sub.1.sup.2 -cM.sub.1.sup.2 is computed by the
arithmetic logic unit 88.
In a completely analogous manner the cavity length control circuit
109 receives a reference signal from the circuit 104 and the AC
part of the intensity sum signal from the summing circuit 65. The
circuit 90 synchronously demodulates the intensity sum and applies
the amplified demodulated DC output in series an in push-pull
fashion with the output from the circuit 104 to the mirrors 18 and
21 so that they move the same distance but in opposite directions
relative to the frame.
It has been shown that the positive scale factor error is
proportional to m.sub.1.sup.2. It has also been shown that one can
effect a closed loop minimization of m.sub.1 which will minimize
this modulation index and correspondingly, the positive scale
factor error.
Since the positive scale factor error is often the dominant scale
factor nonlinearity in both the small angle dithered and rate
biased RLG's, controlling the positive scale factor by closed loop
minimization of m.sub.1 will minimize both the nonlinearity of the
scale factor and the variation of the scale factor error.
FIG. 2 shows a movable mirror and an actuator 122 for dithering the
mirrors 18 and 21 and for providing cavity length control. The
mirror 21, for example, is constrained to translation and is
mounted on a post 123. The periphery of the mirror 21 is
sufficiently thin that it will flex to permit the center of the
mirror to move through a distance adequate to adjust the cavity
length to its optimum value. The actuator 122 is mounted to a frame
134 to translate the post 123. Cavity length control is
accomplished by applying a voltage to the mirror actuator, which
preferably includes a pair of piezoelectric plates 124 and 126
mounted on opposite sides of a thin membrane 128. The voltage
causes one of the piezoelectric plates to contract while the other
expands, which moves the membrane 128, and consequently the mirror
21, in or out with respect to the cavity 22. The structure of the
movable mirror 21 and the actuator 122 are described in U.S. Pat.
No. 4,383,763 issued May 17, 1983 to Hutchings et al. The
disclosure of that patent is hereby incorporated by reference into
this disclosure.
THEORY OF OPERATION
This following description of the operation of the invention
concentrates on computing the intensity and frequency variation due
to retroscatter in the ring laser gyro. It shows explicitly how to
incorporate the resonator leg lengths and the mirror scatter
asymmetry into the phasor addition of the scatter from the mirrors.
It takes into account the two orthogonal types of scatter, either
that due to dielectric variation or that due to height variation,
which lead to common mode phase shifts of 0 and .pi./2 radians,
respectively. It also computes the lockin rate and scale factor
errors in terms of experimentally observable quantities. The
computations are done to first order in intensity variation which
permits us to include some second order frequency pulling terms
which are needed for the scale factor corrections.
The retroscatter from mirrors in the RLG induces frequency and
intensity variations of the individual modes. When the frequency
variation (which is sinusoidal with a frequency equal to the beat
frequency) of the two modes of the RLG becomes such that the
frequencies are equal for a fraction of a beat period, the
frequencies become locked and the RLG can no longer measure
rotation. This is the condition called lockin. In a popular
implementation of the RLG, it is largely subdued by placing a
sinusoidal plus random mechanical rotational dither of about 200
arc seconds amplitude and at a rate of a few hundred Hertz on the
input axis of the RLG. This greatly reduces the lockin rate but
still results in a significant random drift due mainly to our
inability to accurately control the amplitude of the dither. It is
generally this residual random drift that is minimized in attempts
to reduce the scatter by push-pull control (.pi. phasing) of two
mirror positions.
The frequency variation (often called frequency pulling) is
influenced by the intensity variation as well as by the scatter
magnitude, so in order to obtain a complete picture of the lockin
and scale factor effects of the scatter, requires solving the
intensity equations as well as the frequency equations.
Mirror scattering due to surface roughness or height variation, H,
induces a common mode phase delay of 90 degrees or .pi./2 radians.
Mirror scattering due to dielectric variations, D, induces a phase
delay of 0 degrees. These two scattering sources also have
asymmetrical phase shifts which result in the clockwise (cw)
traveling scattered field appearing as the complex conjugate of the
counter-clockwise (ccw) scattered field. It is this asymmetric
phase shift that is being varied when mirror .pi. phasing is
performed. The retroscatter looks like a sinusoidally time varying
electromagnetic energy loss per pass, l, to the primary waves. It
is found that l.sub.H.sup.+ =-l.sub.H.sup.- and l.sub.D.sup.+
=l.sub.D.sup.- where the superscripts + and - refer to cw and ccw
propagating scattered light.
The intensity equations from Aronowitz et al., "Positive Scale
Factor Correction in the Laser Gyro", IEEE J. Quantum Electronics
QE-13, 338 (1977) can be written in terms of these scattering
losses: ##EQU1## where .alpha. is the small signal excess gain per
pass, .beta. and .theta. are the saturation and cross-saturation
terms, respectively, and I.sup.+ and I.sup.- are the cw and ccw
beam (mode) powers respectively.
Rewriting Equations 1 and 2 by taking the sums and differences of
the beam powers gives: ##EQU2## where l.sub.H and l.sub.D are the
time dependent loss terms associated with H and D scatter and
##EQU3##
These equations may be further simplified by writing the intensity
sum as a combination of the time average value of the sum ##EQU4##
and its time dependent part, which is denoted I.sub.V. When this is
done, Equations 3 and 4 may be rewritten omitting terms squared in
i or I.sub.V since these are very small compared to I.sub.o.sup.2
and also drop terms in li and lI.sub.V since these are also very
small ##EQU5##
Equation (5) is provided and solved in Aronowitz et al., "Positive
Scale Factor Correction in the Laser Gyro", IEEE J. Quantum
Electronics QE-13, 338 (1970). Equation 6 is of the same type as
Equation (5). Since l.sub.H and l.sub.D are periodic at the beat
frequency, .omega., the solutions when the rotation rate is not too
near the lockin rate are:
and
where t stands for time and t, and t.sub.2 are phase delays that
will be defined shortly.
It may be shown that: ##EQU6## where the E's are now the
time-dependent magnitudes of the mode fields.
Incorporating Equations (10') and (10") yields: ##EQU7## where
t.sub.1 and t.sub.2 are the inverse of the delays of the laser mode
power response to step changes in the mode loss differences and
sums, respectively.
H and D are the fractional magnitudes of the H and D scattered
fields and .phi..sub.H and .phi..sub.D are the asymmetry phases of
these scattered fields. These phases are varied in synchronism for
a given mirror during mirror .pi.-phasing since both the H and D
type scatterers are fixed to that mirror. However, when combined
with the scatter from other mirrors the variation with voltage of
.phi..sub.H and .phi..sub.D for the four mirror resultant scatter
will not generally be in synchronism. It is fairly simple to model
this behavior by choosing four single mirror d's and four single
mirror h's and varying the asymmetry phases of two of them as is
done in .pi.-phasing operations. Thus a measurement of .phi..sub.H
is not generally accurate enough to predict the change in
.phi..sub.D and vice versa. In addition, when the magnitudes H and
D are varied during .pi.-phasing operations the minimum H
(difference intensity modulation) will occur at one .pi. phasing
voltage and the minimum D (sum intensity modulation) will generally
occur at a completely different voltage. Since these are not
generally at the same voltage, the minimum I.sup.+ is not expected
to occur at the same voltage as the minimum I.sup.-. It has been
shown that .beta. and .tau. are nearly equal for our operating
parameters (pressure, gain to loss ratio, etc.) so that
.omega..sub.g <<.omega..sub.a. Then at low rotation rates
where .omega.<.omega..sub.g gives i.sub.1 >>I.sub.1 even
when H=D and generally it is expected H>>D if our mirror
films are non-crystalline (amorphous) and there are no large
particles contributing to retroscatter. Thus it will be difficult
to measure I.sub.1 but, as shall be seen subsequently, its
measurement is the only convenient gauge of the magnitude and phase
of the main frequency pulling term.
The cw and ccw intensities may now be written in terms of our first
order solutions for them:
and these are the final results for the intensity modulation
equations to first order.
Equations (15) and (16) permit comparison of the relative apparent
phases of I.sup.+ and I.sup.-. Even for equal H and D (usually
H>>D), i.sub.1 exceeds I.sub.1 by about a factor of 10 due to
the large value of ##EQU8## Thus twice the average scatter phase,
2.epsilon., will usually appear to be 180 degrees because the
intensity modulations are pulled to this value by the large ratio
of .omega..sub.a /.omega..sub.g. This is a consequence of mode
competition. When the mode losses become unequal as happens during
the beat cycle, the mode with the lowest losses quickly (with a
time constant of .omega..sub.g.sup.-1) "grabs" more than its share
of atomic gain. This competition is enhanced at higher pressures
because the pressure broadening causes hole overlap even with two
neon isotopes.
The first order iterative result for the frequency modulation in
the RLG are obtained using the same retroscatter phase and
magnitude parameters that were used in the previous section, namely
H, D, .phi..sub.H, and .phi..sub.D. It has been shown that:
##EQU9## The frequencies of the beams including the retroscatter
terms are obtained using the Equations 17' and 17" by dropping the
second order terms in D and H since they are much less than 1:
##EQU10## where .omega..sub.o is the empty resonator frequency
without rotation and .omega..sub.i is the geometrically defined
frequency splitting (8.pi.A.OMEGA./.lambda.L) of the waves due to
inertial rotation rates. .phi..sub.b is the instantaneous phase
difference between the two primary waves. The quantity of interest
is the difference between .omega..sup.+ and .omega..sup.- since
this is a quantity which permits us to obtain the intensity versus
rotation rate when the beams are mixed on the heterodyne
photodetector, which gives the rotation rate and lockin rate. The
difference frequency is obtained by subtracting Equation (17b) from
(17a) and is: ##EQU11## where it is not difficult to show that
E.sup.- E.sup.+ =(I.sup.2 -i.sup.2).sup.1/2 so that the first
intensity term in braces is 2 if terms of order ##EQU12## can be
neglected. To the same order of approximation, the second intensity
term can be written ##EQU13## is the modulation index of the
intensity difference. Here rot has been used interchangeably with
.phi..sub.b in the small scattering terms only since at the level
of iterative approximation given here this will be sufficiently
accurate. Then with this notation ##EQU14##
The function can be expanded with t.sub.1 in its argument and to
yield terms which are constant in time and terms which are varying
at .omega. and 2.omega.: ##EQU15##
The constant term is part of the scale factor error called the
"positive scale factor error". The term which oscillates at .omega.
is usually the main term responsible for lockin since it is first
order in the retroscatter coefficient, D. The terms at 2.omega. can
also result in lockin but usually at a lower rate than that due to
ones which vary at .omega.. By modeling the .omega. and 2.omega.
terms with many combinations of .phi..sub.D and .phi..sub.H and and
D and H, it can be shown that with an error of less than 20 percent
even when H is large the lockin rate is given by ##EQU16## which
explicitly shows the dependence of lockin rate on D and H.
Equation (21) may be simplified using ##EQU17## which are
measurable experimentally as the maximum modulation indices of the
difference and sum intensities, respectively which are seen at
input rates lower than .omega..sub.a and .omega..sub.g,
respectively. Also expressed in modulation indices is ##EQU18##
Experimentally, .omega..sub.a and .omega..sub.g may be determined
as the half width at 0.7 maximum of the M.sub.1 and m.sub.1
intensity modulation indices as a function of rate as seen from
Equations (11) and (12). M.sub.2 and m.sub.2 are the maximum values
of the plots of M.sub.1 and m.sub.1.
If ##EQU19## as is usually the case, then one minimizes M.sub.2 to
minimize lockin rate. Since m.sub.2, .omega..sub.a, M.sub.2 and
.omega..sub.g are conveniently available in a dithered gyro, one
can always minimize lockin rate using appropriate control loops to
control M.sub.2 and m.sub.2.
The positive retroscatter scale factor error term is ##EQU20## The
negative scale factor error term is ##EQU21##
The average rotation rate <.omega.> may be calculated as
The lockin rate term is given in terms of the same experimentally
observable parameters defined by Equation (22). The sinusoidal
terms on the right side of Equation (20) lead to a net increase of
the average beat cycle period and therefore cause a negative scale
factor error. This can be seen by direct numerical integration of
Equation (20) when .omega..sub.i is considerably greater than the
lockin rate. The average value of w, the output fringe rate, is
defined as 2.pi. times the inverse of the time required for the
fringes to complete a cycle ##EQU22## Since it is assumed that
.omega..sub.D, .omega..sub.H <<.omega..sub.i, the denominator
is expanded in a Taylor's series to 4th order in .omega..sub.D and
.omega..sub.H so that all of the sine terms are now in the
numerator. Note that ##EQU23## which means that ##EQU24## This
makes the integral easy to do and results in the expression
##EQU25## Dropping higher order terms gives: ##EQU26## so that in
the expression m.sub.1.sup.2 -CM.sub.1.sup.2, the quantity C may be
expressed as ##EQU27## It is very important that C is a function of
.omega..sub.i. The arithmetic logic unit 86 has access to
.omega..sub.i so that it can form the discriminant which includes
both positive and negative scale factor errors.
Numerical integration of Equation (24) and comparison with Equation
(25A) shows that the fractional error in computing the scale factor
error in Equation (25A) is less than 1 percent for .omega..sub.D,
.omega..sub.H '<0.1.omega..sub.i and less than 6 percent for
.omega..sub.D, .omega..sub.H '<0.2.omega..sub.i. What is more
important is that the dependence of Equation (25A) on the phase
difference between the first and second harmonic scatter pulling
terms, 2.phi..sub.H -2.phi..sub.D +t.sub.1, tracks the numerical
result much more closely than the errors stated above. Equation 25
is a departure from the previous expressions which just used the
computed value of the lockin rate as the frequency reducing term
and took no account of the relative phasing between the first and
second harmonic terms in Equation (20).
Note also that the term with 3/8 as the numerical factor has -2 as
its exponent in .omega..sub.i, the input rate. This means that this
term is not really a scale factor error at all but instead a rate
dependent bias since it doesn't change signs when the direction of
rotation is reversed. For an undithered 28 cm gyro rotating at
.omega..sub.i =1.degree./sec with .omega..sub.D =0.1.degree./sec,
.omega..sub.H '=0.1.degree./sec, one can get as high as
(3/8)(0.1/1).sup.2 (0.01/1)3600=0.375.times.10.sup.-2 =10.sup.-2
.times.3600=0.135.degree./hr bias due to this term. This bias is
expected to be highly variable with temperature since it depends on
the phasor-addition-dependent retroscatter magnitudes and is also
dependent on the scatter phases, 2.phi..sub.H -2.phi..sub.D, in the
sine term. However, this bias decreases as .omega..sub.i.sup.-2 as
input rate is increased. The reduction of this bias is another
important point (besides scale factor error) in favor of high input
rate body dither with large random component.
Since generally .omega.'.sub.H <<.omega..sub.D,
.omega.'.sub.H may be neglected in the following. Determining the
input rate, .omega..sub.C, at which the scale factor error is
expected to cross zero involves solving Equation (25) for that
rate, which yields ##EQU28## where terms of order D/a and H/a have
been neglected relative to unity. Thus the rate at which scale
factor error changes sign depends critically on the ratio of D to H
and will be infinite when D>H. This latter result agrees with
Reference 1 where the intensity and frequency Equations were solved
numerically.
Similarly, obtaining the rate, .omega..sub.o slope, at which the
scale factor error has zero slope (and the maximum positive value)
requires differentiating Equation (25) with respect to input rate
and setting this result equal to zero, which gives ##EQU29## The
maximum (positive) error is then given by ##EQU30## and is usually
on the order of 100 ppm on a low loss, low scatter 30 centimeter
gyro and occurs in the vicinity of 3 degrees per second,
(.apprxeq.6000 Hz).
The expected time variation of the heterodyne detector signal,
I.sub.h, taking into account the time variation of the two field
amplitudes due to retroscatter may be expressed as: ##EQU31## where
terms of order (i/I).sup.2 have been dropped. The value of
.phi..sub.b in the second factor of Equation (32) assumes zero
differential path in the beam combiner and that the detector is in
the center of the combined mode pattern. It has also been assumed
that the beam combiner provides equal throughput for both beams.
The result indicates that a significant second harmonic will appear
on the beat (heterodyne) signal even at high input rates where
.phi..sub.b -.phi..sub.i t is a good approximation.
* * * * *